Bifurcation Diagram Bouncing Ball

[bkercso]

Bifurcation map of a bouncing ball on opposing slopes.

Schematic of the system (not my work):


This fractal is similar to bifurcation maps of double pendulum, but calculation is much faster, because this bouncing ball system is not feedbacked (no air resistance etc.), so we do not have to use successive approximation to calculate bouncings: every iteration gives us a point in the picture. This reduces calculation time from weeks to minites.

x-axis: initial velocity perpendicular with the slope
y-axis: velocities parallel to the slopes when bounce
Slopes angle: 60°, initial parallel velocity: 0.2
Equations are from a Hungarian book: Tél Tamás, Gruiz Márton: Kaotikus dinamika, Nemzeti Tankönyvkiadó, 2002.
(http://ttk.nyme.hu/mfi/Documents/Fizika-Kaotikus%20jelens%C3%A9gek/Kaotikus_jelens%C3%A9gek_I.pdf)

Img #1

[cKleinhuis]

this is cool, so, for clarification, what does this image tells us ?
 
and some questions:

- the bifurcation seems not to be visible, since bifurcation theory means splitting in 2, at which zoom levels can you spot the start and ends of the bifurcation
- can you provide zooms to the start of the bifurcation ?
- what do we see here? chaos or order ? i mean, is it chaotical behaviour of the bounces or does it follow certain lines ?

[bkercso]

Thank you for this questions, it is important indeed.
This is the whole phase space of this system (cannot zoom out), so we can say this do not produces classical bifurcations. I call this picture bifurcation map because it is generated the same way as classical bifurcation maps. This system as is: not have periodic sections. In the other topic (http://www.fractalforums.com/new-theories-and-research/bifurcations-fractals-discovery/) I posted an explainig picture about that:


So this system has only quasy periodic- and chaotic regions.

[cKleinhuis]

i see, it makes perfect sense to analyse existing dynamic systems like that

the interesting parts for these diagrams are the changing regions, can you render a zoom into a spike of the periodic line ?
or to the chaotic areas?

[bkercso]

In fact I did that, I will found a good picture upload site for uotflank daily limit of this site.
It is a fractal, so you can see the same things in zooms too: only bubbles, namely quasy periodicity. In an other render shows not perpendicular but parallel velocity when bounce you can see interesting spikes...
I think the stucture of this fractal is similar with double pendumum's, but because of faster computation I can make deeper zooms. We will see if structure complexity is higher in zooms or not.

[bkercso]

I played a little with this system and find that if initial velocity is perpendicular to slope that system has periodic behavior (with chaotic regions of course), else it is quasy peiodic (+ chaotic).

I think light patterns come from first slope and dark patterns from the other, but I will check it..

Img #2

upload a picture

Video #1
https://youtu.be/-LrV-KsPTu8

[bkercso]

First I zoom the periodic fractal.

This is from clean periodic region, 370x zoom, 5 hours of calculation time.

Img #3
Set:


Zoom:

[bkercso]

And one from the edge of periodic and chaotic regions. 500x zoom, 1.5 hours calculation.

Img #4
Set:


Zoom:


Light green lines in the chaotic region are periodic/soft chaotic regions (?).

[bkercso]

As I see this system's fractals are much more simpler than double pendulum's ones, but this is a good model system this type of bifurcation maps because it requires 1E4 times less computation.

[cKleinhuis]

yay, the underlying method is far simpler, and the "chaotic" or "dense" regions dont exhibit the beautiful sub branching
but the quasiperiodic regions are what we encounter when watching a ball bounce this kind of thing and assume it stays at a certain position longer, or we seem to be able to exactly predict the movements of the ball ( more or less )

[Chillheimer]

thank you bkercso.
I love what you do!
It's very inspiring!
 


I know this is a bit off-topic, but I've found a huge interest in bifurcation diagrams lately.
I believe that bifurcation and especially the "periodic order windows" that occur out of chaos are hugely important on a larger scale in reality.

I totally agree with the ideas of Robert Paterson:
http://smartpei.typepad.com/robert_patersons_weblog/2009/04/the-fractal-history-of-mankind-part-2-our-time.html

ray, kurzweil, singularity is near - nice read!

and actually I believe that: (see attached image)

[bkercso]

cKleinhuis:
Well, now I think there is no contoured sub branching in chaotic regions. I made deep zooms on it and it's dimness I cannot attributed to rough time resolution because there is no time resolution here.
I see shaped patterns in it only when I take less iterations.

[bkercso]

Chillheimer:
Thank you.

There is no doubt that our global word is unsustainable. Not only because of oil and wasting, but the picture is quite complex and on many areas there are majore problems, which has own fractal-like fine structure. Well, only the history we can see clear (if we)...

[bkercso]

Img #5

Set:



Zoom 78,000x (made by 15,000,000,000 bounces, during 35 minutes):

[bkercso]

I recalculated Img #4 with higher point/pixel density and upload to forum's gallery.